Find Entire Functions for f(2-i)=4i and |f(z)|≤e^2

[jaci55555]

find all the entire functions f for which f(2-i) = 4i and |f(z)|<= e^2


if f is entire on Complex, and there exists M,K element R^+ and k element N such that
|f(z)|<= M * |z^k|
for all z element Complex with |z|>=K then f is polynomial with degree <=k


can i use this thereom? if i do, then the |z^k| part is 1 and so M is e^2.
but then i should try find the k...
I'm not sure

i know that x^2 - 6X + 9 is an entire function there, but i just worked it out...

 

[micromass]

find all the entire functions f for which f(2-i) = 4i and |f(z)|<= e^2

Do you mean that this holds for all z?
Do you know Liouville's theorem?

 

[snipez90]

The OP gave a stronger theorem than Liouville, which is the case k = 0 in the above theorem.

 

[micromass]

The OP gave a stronger theorem than Liouville, which is the case k = 0 in the above theorem.

Yes, I'm aware. But Liouville is enough here. Stronger theorems somethimes obfusciate things...

 

[jaci55555]

Do you mean that this holds for all z?
Do you know Liouville's theorem?

Thanks for answering :)
That is all the question stated - so i think it does hold for all z.
I thought the theorem i stated above WAS Liouville's theorem?
But i wasn't sure about how to use it.

 

[micromass]

Yes, of course the theorem you stated is Liouville's theorem. But the theorem originally known as Liouville's is "every entire, bounded function is constant". This follows easily from the theorem you stated (try to prove this!)

 

[jaci55555]

thank you